Bottom Line:
We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected.We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph.Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

ABSTRACTWe consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

Fig1: (a) An undirected tree indicating two nodes which we infect to initiate epidemics, with all other nodes initially susceptible. (b) The mean (dots) of 100,000 stochastic simulations on the network with transmission rate τ=0.1 across each link and removal rate γ=0.05 for each node, with error bars denoting the 5th and 95th percentiles plotted together with the solution of (3) (solid line) using the Matlab code published with Sharkey (2011)

Mentions:
Figure 1 shows the numerical solution of (3) for a small network of 9 nodes where it is clear that it is accurate to within the precision visible on the graph. Matlab code for solving the system of Eqs. (3) is provided in Sharkey (2011). This code also works on networks which are not trees but is no longer exact in these cases. Cycles in the underlying graph of order three utilise the alternative closure 〈AiBjCk〉=〈AiBj〉〈BjCk〉〈AiCk〉/〈Ai〉〈Bj〉〈Ck〉 which is believed to gain increased accuracy in most circumstances, but these do not occur in the tree graphs considered in the present work. Fig. 1

Fig1: (a) An undirected tree indicating two nodes which we infect to initiate epidemics, with all other nodes initially susceptible. (b) The mean (dots) of 100,000 stochastic simulations on the network with transmission rate τ=0.1 across each link and removal rate γ=0.05 for each node, with error bars denoting the 5th and 95th percentiles plotted together with the solution of (3) (solid line) using the Matlab code published with Sharkey (2011)

Mentions:
Figure 1 shows the numerical solution of (3) for a small network of 9 nodes where it is clear that it is accurate to within the precision visible on the graph. Matlab code for solving the system of Eqs. (3) is provided in Sharkey (2011). This code also works on networks which are not trees but is no longer exact in these cases. Cycles in the underlying graph of order three utilise the alternative closure 〈AiBjCk〉=〈AiBj〉〈BjCk〉〈AiCk〉/〈Ai〉〈Bj〉〈Ck〉 which is believed to gain increased accuracy in most circumstances, but these do not occur in the tree graphs considered in the present work. Fig. 1

Bottom Line:
We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected.We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph.Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

ABSTRACTWe consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.